I propose the "2/3 rule" for multi-factor authentication: to access a system, a user must pass at least two out of three possible methods for authentication. Unlike standard multi-factor authentication, this strategy protects against both false positives (granting access to a hacker) and false negatives (denying access to a legitimate user).

Multi-factor authentication

When you sign into an online service, you're required to provide a username to identify who you are and a password to prove that you are who you say you are. Anyone can claim that they're romanhn on GitHub, but only I know my GitHub password. But if a hacker breaches GitHub's servers and publishes a list of everyone's password,^[1] knowing my password is no longer proof of anything. This is the motivation behind two- or multi-factor authentication (MFA): to access my GitHub account, not only would you have to know my password, but you'd need to authenticate yourself using an additional factor, like tapping a notification on my phone.

Adapted from sketchplanations

I'll use the "Swiss cheese model", shown above, to explain how multi-factor authentication provides security through redundancy. Even if a hacker can pass through one hole (discover my password), they still have to find a hole in another layer (get me to tap the notification) in order to hack my account.

Categories of factors

A user can use any of three main categories of factors to authenticate themself:^[2]

1. A piece of information only they know, like a password or PIN.
2. A physical object only they possess, like a key or security token.
3. A physical characteristic only they display, like a fingerprint or voice pattern.

Each layer should be a different category because different factors in the same category are often breached simultaneously, as if they were sliced from the same section of the cheese block so their holes were in similar places. For example, if I fell victim to a phishing attack, the hacker would discover all the "pieces of information only I know" in one fell swoop. Asking the hacker "what was the name of your first pet?" doesn't provide any additional security if they discovered my password and my first pet's name at the same time.

This is the same idea behind the "3-2-1 rule" for backing up data:

You should maintain three copies of anything you care about: two copies in different formats and one off-site backup.

Like with authentication, redundancy in backups is best when it is spread across different modalities. Instead of protecting against a phishing attack, however, it prevents a house fire from wiping out your baby photo collection because you never stored a copy in the cloud.

The problem with MFA

While MFA makes it harder for a hacker to breach an account, it also makes it easier for a legitimate user to get locked out. People lose their cell phones and biometric sensors fail, all the time. And this matters because getting locked out is often worse than your account being hacked – I would rather let a hacker download all of the data in my Google Photos account than lose access to it permanently.

Suppose that a laptop can be accessed using three authentication factors (entering a password, confirming via an SMS text, and scanning a fingerprint), and they each have false positive (hacker successfully bypasses) and false negative (a user fails to pass) rates of 1%. Different multi-factor authentication schemes combining these factors have new rates, as shown by the confusion matrices below.^[3]

Only password required

user	access granted	access denied
legitimate	99.00%	1.00%
hacker	1.00%	99.00%

Password and SMS required

user	access granted	access denied
legitimate	98.01%	1.99%
hacker	0.01%	99.99%

Password, SMS, and fingerprint required

user	access granted	access denied
legitimate	~97.03%	~2.97%
hacker	0.0001%	99.9999%

With each additional factor, the chance that a hacker could successfully break in decreases exponentially (1% -> 0.01% -> 0.001%). However, the chance that a legitimate user gets locked out increases ~linearly (1% -> 1.99% -> ~2.97%). The probability that "something bad happens" is actually greater when three factors are required instead of two.^[4]

The 2/3 rule

To combat against false negatives and false positives simultaneously, I propose the "2/3 rule" for multi-factor authentication:

A system should have in place three methods, in different modalities, for authenticating a legitimate user. The user is granted access to the system as long as they can pass at least two out of the three methods.^[5]

This provides redundancy on both fronts: a malicious actor must crack at least two methods to breach an account, and a legitimate user must fail at least two methods to be locked out. It's like a castle with a moat and cannons to thwart invaders, as well as guardrails and nets to prevent the king from falling into the moat.

The confusion matrix for this combination demonstrates how the 2/3 strategy provides symmetric redundancy:

At least two of password, TOTP, or fingerprint

user	access granted	access denied
legitimate	~99.97%	~0.03%
hacker	~0.03%	~99.97%

While the chance that a hacker breaks into a 2/3 system is higher than the original 2- or 3-factor systems (~0.03% > 0.01% > 0.0001%), the chance that a user gets locked out is dramatically lower, even more so than the 1-factor system (~0.03% < 1.00%). This means that the overall chance of being hacked or locked out is lower than any other system.

This is nothing new

That being said, this strategy is kind of already in use; I just wanted to give a name to it. For example, if you have 2-factor authentication enabled on your Google or Facebook accounts, they will allow you to receive a 2FA code from a text message in case you can't access your TOTP generator.^[6]

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1. Okay, a more realistic situation is "a hacker breaches GitHub's servers, downloads a list of everyone's password hashes, cracks the easiest ones, and sells the list to another hacker on some forum", but the same idea holds. ↩︎

2. There are other categories like "somewhere they are", which can be measured by their IP address at the time of login. ↩︎

3. The probability that a legitimate user is denied access is $1-{0.99}^n$ , where $n$ is the number of factors required. Similarly, the probability that a hacker is granted access is ${0.01}^n$ . This model becomes more complicated if different factors have different false positive/negative rates or are combined in different ways (for example, one scheme could require either a password or a fingerprint). ↩︎

4. This explicitly assumes that any given login attempt has an equal chance of being malicious or legitimate, and it implicitly assumes that the "hacked" and "locked out" events are equally bad. Neither of these assumptions are true for most systems. ↩︎

5. Using the fourth category of authentication factors "location", one could implement a "2/4" or "3/4" strategy. ↩︎

6. Technically, using a TOTP generator and receiving a text to your phone both fall under the "something you have" category. A better example would be if you could use a fingerprint instead of a 2FA code. ↩︎